DLR Equations and Rigidity for the Sine‐Beta Process

Abstract

We investigate Sineβ, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one‐dimensional log‐gases, or β‐ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of Sineβ using the Dobrushin‐Lanford‐Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sineβ to a compact set, conditionally on the exterior configuration, reads as a Gibbs measure given by a finite log‐gas in a potential generated by the exterior configuration. In short, Sineβ is a natural infinite Gibbs measure at inverse temperature β > 0 associated with the logarithmic pair potential interaction. Moreover, we show that Sineβ is number‐rigid and tolerant in the sense of Ghosh‐Peres; i.e., the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long‐range interactions in arbitrary dimension. © 2020 Wiley Periodicals, Inc.

Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

“…The rigidity property for Sine β in the sense of Gosh-Peres was proved by Chhaibi-Najnudel [12] and Holcomb-Paquette [22] computed the leading order of the maximum eigenvalue counting function. Finally, Leblé [38] gave recently an alternate proof of Theorem 1.12 for test functions of class C 4 c (R) which relies on the DLR equations for the Sine β process established by Dereudre-Hardy-Leblé-Maïda [16].…”

The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

“…Combining (5.27) and (5.31), we obtain (for R large, the constant error term in (5.27) can be absorbed in the dominant error terms) 5 2 jlog js: First, we observe that the error term j log js is of the same order as ErrScr and s was chosen small enough so that ErrScr is small. On the other hand the limit defining the specific relative entropy is non-decreasing (it follows from a superadditivity argument, see e.g., [9,Cor.…”

“…In [5], a different description of Sine is given using the Dobrushin-Landford-Ruelle (DLR) formalism, but the question of whether Sine is the unique solution to DLR equations is left open. The main result of the present paper answers positively to a slightly different uniqueness question, phrased in terms of the log-gas free energy.…”

Number-Rigidity and $β$-Circular Riesz gas

“…This property has been introduced in [14] and studied for a large variety of point processes as for instance the Gaussian zeros and Ginibre process [11], perturbed lattices [21], stable matchings [16] or Pfaffian point processes [2]. The number rigidity of the Sine β process for any inverse temperature β > 0 has been proved independently in [4] and [23] with two drastically different approaches. In [23] the authors follow the strategy of [11] which consists of controlling the variance of linear statistics whereas in [4] the authors lean on a statistical physics approach which has inspired the present work.…”

“…We follow a statistical physics approach based on the canonical DLR equations. It is inspired by the recent paper [4] where the authors prove the number-rigidity of the Sine β process.…”